A new type of approximation for the radical quintic functional equation in non-Archimedean (2,β)-Banach spaces

Let R be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean (2,β)-normed spaces (X,‖⋅,⋅‖⁎,β) and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equationf(x5+y55)=f(x)+f(y),x,y∈R. Also, under some weak natural assumptions on the function γ:R×R×X→[0,∞), we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when f:R→X satisfy the following radical quintic inequality‖f(x5+y55)−f(x)−f(y),z‖⁎,β≤γ(x,y,z),x,y∈R∖{0},z∈X, with x≠−y.